Nano-electromechanical spatial light modulator enabled by asymmetric resonant dielectric metasurfaces

Spatial light modulators (SLMs) play essential roles in various free-space optical technologies, offering spatio-temporal control of amplitude, phase, or polarization of light. Beyond conventional SLMs based on liquid crystals or microelectromechanical systems, active metasurfaces are considered as promising SLM platforms because they could simultaneously provide high-speed and small pixel size. However, the active metasurfaces reported so far have achieved either limited phase modulation or low efficiency. Here, we propose nano-electromechanically tunable asymmetric dielectric metasurfaces as a platform for reflective SLMs. Exploiting the strong asymmetric radiation of perturbed high-order Mie resonances, the metasurfaces experimentally achieve a phase-shift close to 290∘, over 50% reflectivity, and a wavelength-scale pixel size. Electrical control of diffraction patterns is also achieved by displacing the Mie resonators using nano-electro-mechanical forces. This work paves the ways for future exploration of the asymmetric metasurfaces and for their application to the next-generation SLMs.


RESONANT DIELECTRIC METASURFACES
For resonant metasurfaces or photonic crystals, it is known that the temporal response of the resonator can be described by temporal coupled mode theory (TCMT) [1,2]. As seen in Fig. 1b in the main text, with normally incident light, the resonant metasurface can be modeled by a single-mode resonator that is coupled to two ports. The dynamics of the optical resonance can be generally formulated by: where and 0 correspond to complex amplitude of the resonance and the central resonance frequency, respectively; 1 and 2 are the coupling coefficients between the two ports and the resonances; the resonance radiatively decays into port 1 and 2 with decay rates of 1 1 and 1 2 , respectively; 1 is the nonradiative decay rate; + 1 ( − 1 ) and + 2 ( − 2 ) are amplitudes of the incoming (outgoing) waves from the ports; is a direct-transport scattering matrix, written by = , where , , and are the real reflection coefficient, the real transmission coefficient, and the phase factor, respectively. generally describes the direct coupling between the incoming and outgoing waves.
In addition, as depends on the selection of reference planes in the model, can be set to 0 for simplicity. In addition, we here assume that 1 is negligible because silicon is almost lossless in the telecom wavelength range. According to the time-reversal symmetry and the energy conservation, the coupling coefficients satisfy * revealing that the coupling conditions are fundamentally related to the decay rates of the resonances as well as the direct-transport scattering [2,3].
According to Refs. [3,4], we could analytically solve Eqs. S1-S4 to obtain the Eqs. 1 and 2 in the main text. In detail, when the system is driven by a continuous laser, whose frequency is , we can derive as a function of from Eq. S1 By inserting Eq. S5 into Eq. S2, the outgoing waves can be described by, From Eq. S6, we can derive the reflection spectra of port 1 and 2, 1 and 2 : Next, Eqs. S3 and S4 are employed to eliminate phase ambiguity of 1 and 2 in Eqs. S7 and S8.
Specifically, from Eq. S3, 1 and 2 can be described by where 1 and 2 are phases of the coupling coefficients of 1 and 2 , respectively. By inserting Eq. S9 into Eq. S4, (2 1 ) and (2 2 ) can be derived by Then, (2 1 ) and (2 2 ) are expressed as: (S13) As 1 and 2 in Eqs. S7 and S8 can be described by cos(2 1 ), cos(2 2 ), sin(2 1 ), and sin(2 2 ), we can derive the Eqs. 1 and 2 in the main text: (S15) The two-port resonator model shown in Fig. 1b in the main text generally describes any singlemode resonant metasurfaces under normal incidence. When the light is obliquely incident and the metasurface is in sub-wavelength regime (i.e. there is no diffraction), the metasurface can be modeled by a four-port resonator. The general description of the four-port resonator model can be found in Ref. [4]. Here, we only deal with a fully symmetric case where the resonance equally decays into the four ports with the decay rate of 1 0 . The reflection spectrum of the symmetric resonator, , can be expressed as [4]: (S16) We should note that Eq. S16 becomes identical to Eq. S14 or S15 when 1 1 = 1 2 = 1 0 . In other words, if the structures are symmetric with respect to all available ports and 1 0 ≠ 0, the single-mode resonance is always critically coupled to the excitation.

NANT DIELECTRIC METASURFACES
To physically implement a symmetrical case of the theoretical model in Supplementary Note 1, we simulate the metasurfaces possessing the mirror symmetry in the -direction. In Fig. S4a, the metasurface grating is composed of Si nanobars and surrounded by air. Specifically, the 841 nm wide and 838nm thick 2D nanostructures are periodically arranged with the lattice constant of 1093 nm. Figure S4b shows an electrical field profile of the TE-polarized eigenmode at Γ point.
The mode in Fig. S4b originates from the Mie mode hosted by individual Si nanostructures (see and S4d agree with the previous discussion of the symmetric resonator in Supplementary Note 1.

SUPPLEMENTARY NOTE3: NUMERICAL INVESTIGATIONS ON BEAM STEERING OF THE ASYMMETRIC METASURFACES
Here, we numerically verify the metasurfaces' capability of beam steering, using a pair of the nanostructure as a building block of the proposed active metasurfaces. Specifically, the gaps of the pairs of nanostructure are adjusted by the applied biases such that the metasurface manipulates the wavefronts of the reflected light. When assuming that the phase is locally determined by the gap of the two nanostructures, we can exploit the relationship between 0 ℎ and 1 − 2 2 plotted in Fig. 2d in the main text as a lookup table to design the metasurfaces. In other words, once the desired phase distribution is determined, the gaps of nanostructures can be inversely obtained from Fig. 2d in the main text. It is noteworthy to mention that this lookup table approach is widely used in passive and active metasurfaces. First, we investigate a blazed diffraction grating of which the linear phase gradient is negative. As shown in Fig. S6a, the period of the grating, , is determined by periodicity and 2Λ, where the periodicity represents the number of the pairs in one period of the blazed grating. can be also found in the case of periodicity of 1 (see Supplementary Figure 5 for details). We expect that these differences inherently result from the asymmetry of the structure with respect to -axis.
Besides, at the design wavelength of 1529 nm, the reflected power coefficients of all available diffraction orders are plotted in Fig. S7, showing that all high-order diffraction components are suppressed compared to the desired diffraction order. In particular, it is worth noting that the highest-order diffractions at the angle ±44 • are well suppressed when the periodicity is extended over 1.  Figs. 1e and 1f in the main text is fitted by using Eqs. 1 and 2 in the main text. As shown in Fig. 1c in the main text, | 1 | 2 (| | 2 ) and 1 ( 2 ) represent reflection and reflected phase for top (bottom) illumination, respectively. From the fitting, we find the Q-factor and 2 1 of 1004 and 17.52, respectively. a: Calculated and fitted spectra of | 1 | 2 and | 2 | 2 . Red asterisks show the calculated spectra of | 1 | 2 or | 2 | 2 . The fitted spectrum is plotted by a black solid line. b: Calculated and fitted spectra of 1 and 2 . Red and blue asterisks represent the calculated spectra of 1 and 2 , respectively. The fitted spectra of 1 and 2 are plotted by solid and dashed black lines, respectively. c Fitted reflection spectra of the direct-transport scattering process.   With three pairs of the asymmetric nanostructures grounded, only one pair is connected to external bias, , for every four pairs of nanostructures. As two electrodes are required for this electrical bias, a large number of devices can be fabricated and modulated simultaneously [5]. b Measured reflection spectrum for TE-polarized normally incident light. The spectrum is measured without any bias and normalized by the reflection from a gold electrode. c Measured intensity at the Fourier plane of the metasurface. The intensity is measured at 1519 nm as a function of the applied bias. changes from 0V to 7.6V with a step size of 0.101 V. At each bias, the intensity is normalized by the maximum intensity. d Measured diffraction patterns at 1519 nm with of 7.093V. The corresponding data is noted by a blue dashed line in c. Top: The normalized intensity image is measured at the Fourier plane of the metasurface. The value of is noted on top of the image. Scale bar denotes 0.05 0 where 0 is a magnitude of wave vector in free-space. Bottom: Measured cross-sectional intensity profile is plotted as a function of the diffraction angle. The intensities are normalized by the peak intensity around −10 • . The diffracted signals near ±10 • are denoted by red shades. Quantitatively, the −1st and +1st order signals are 6.05 dB and 3.75 dB larger than the 0th order signal, respectively